Yes to the bolded.
With the dominant wind out of the NW in the winter time over the warm lake water, you're going to have persistent advection of moisture inland that's enough to ensure the reproduction and maintenance of clouds (though not necessarily precip) with just a wee bit of rising air.
Unless you get a strong high pressure system behind a major storm to produce enough sinking air to overwhelm the moisture advection off the lake, or an offshore wind flow, you're shit out of luck if you want to see much sun.
That is (in part) why I'm down south now, lol.